real analysis – Explanation for the “pathological” behavior of the sum of uncountably many nonnegative numbers

Let ${x_lambda}_{lambda in Lambda}$ be a set of (possibly) uncountably many nonnegative real numbers. We know that the sum defined by
$$
sum_{lambda in Lambda} x_lambda :=sup{text{finite partial sums (over finite subsets of $Lambda$)}}
$$

is finite only if all but countably many $x_lambda$ are zero. See this question for a reference.

I find that there are some “uncomfortable” aspects of this definition of sum. It is not a limit of a sequence (although it is the limit of a net). To approximate uncountablility by a finite subset seems too far-fetched.

Moreover, in the case all $x_lambda$ are zero, the sum should be essentially $0times infty.$ To see how this cause troubles, just remember we hardly ever use this uncountable sum in measure theory – we only allow countable additivity in measure theory: $mathbb R,$ uncountable additivity in the sense described above forces every set to have zero measure when all singleton sets have zero measure.

What are the root cause of such failure of an uncountable sum with measures? What is the real reason why, and what excatly is missing?

To be clear, which properties of a countable sum fails for this uncountable sum and causes these problems?

Related: measure theory for regular cardinals, Do sets with positive Lebesgue measure have same cardinality as R?